Andrew's git - notes.git/blob - methods/calculus.tex

   1\documentclass[methods-collated.tex]{subfiles}
   2
   3\begin{document}
   4
   5\section{Calculus}
   6
   7\subsection*{Average rate of change}
   8
   9\[m \operatorname{of} x \in [a,b] = \dfrac{f(b)-f(a)}{b - a} = \frac{dy}{dx}\]
  10
  11\colorbox{cas}{On CAS:} Action \(\rightarrow\) Calculation
  12\(\rightarrow\) \texttt{diff}
  13
  14\subsection*{Average value}
  15
  16\[ f_{\text{avg}} = \dfrac{1}{b-a} \int^b_a f(x) \> dx \]
  17
  18\subsection*{Instantaneous rate of change}
  19
  20\textbf{Secant} - line passing through two points on a curve\\
  21\textbf{Chord} - line segment joining two points on a curve
  22
  23\subsection*{Limit theorems}
  24
  25\begin{enumerate}
  26    \def\labelenumi{\arabic{enumi}.}
  27    \tightlist
  28  \item For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
  29  \item \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
  30  \item \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
  31  \item \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
  32\end{enumerate}
  33
  34A function is continuous if \(L^-=L^+=f(x)\) for all values of \(x\).
  35
  36\subsection*{First principles derivative}
  37
  38\[f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}\]
  39
  40Not differentiable at:
  41\begin{itemize}
  42    \tightlist
  43  \item discontinuous points
  44  \item sharp point/cusp
  45  \item vertical tangents (\(\infty\) gradient)
  46\end{itemize}
  47
  48\subsection*{Tangents \& gradients}
  49
  50\textbf{Tangent line} - defined by \(y=mx+c\) where
  51\(m={dy \over dx}\)\\
  52\textbf{Normal line} - \(\perp\) tangent
  53(\(m_{{tan}} \cdot m_{\operatorname{norm}} = -1\))\\
  54\textbf{Secant} \(={{f(x+h)-f(x)} \over h}\)
  55
  56\begin{cas}
  57  \textbf{In main}: Interactive \(\rightarrow\) Calculation \(\rightarrow\) Line \\
  58  \-\hspace{1em} \texttt{tanLine(f(x), x, p)} \\
  59  \-\hspace{1em} \texttt{normal(f(x), x, p)} \\
  60  where \texttt{p} is the \(x\)-value of the coordinate
  61
  62  \textbf{In graph}: define function, then Analysis \(\rightarrow\) Sketch \(\rightarrow\) (Normal \textbar{} Tan line). Type \(x\) value to solve for a point. Return to show equation for line.
  63\end{cas}
  64
  65\subsection*{Strictly increasing/decreasing}
  66
  67For \(x_2\) and \(x_1\) where \(x_2 > x_1\):
  68
  69\begin{itemize}
  70    \tightlist
  71  \item \textbf{strictly increasing}\\ where \(f(x_2) > f(x_1)\) or \(f^\prime(x)>0\)
  72  \item \textbf{strictly decreasing}\\ where \(f(x_2) < f(x_1)\) or \(f^\prime(x)<0\)
  73  \item Endpoints are included, even where gradient \(=0\)
  74\end{itemize}
  75
  76\subsection*{Stationary points}
  77
  78\begin{align*}
  79  \textbf{Stationary point:} && f^\prime(x) &= 0 \\
  80  \textbf{Point of inflection:} && f^{\prime\prime} &= 0
  81\end{align*}
  82
  83\begin{tikzpicture}
  84  \begin{axis}[xmin=-21, xmax=21, ymax=1400, ymin=-1000, ticks=none, axis lines=middle]
  85    \addplot[color=red, smooth, thick] gnuplot [domain=-15:15,unbounded coords=jump,samples=500] {x^3-3*x^2-144*x+432} node [black, pos=1, right] {\(f(x)\)};
  86    \addplot[color=darkgray, dashed, smooth, thick] gnuplot [domain=-15:15,unbounded coords=jump,samples=500] {3*x^2-6*x-144} node [black, pos=1, right] {\(f^\prime(x)\)};
  87    \addplot[mark=*, blue] coordinates {(1,286)} node[above right, align=left, font=\footnotesize]{inflection \\ (falling)} ;
  88    \addplot[mark=*, orange] coordinates {(-6,972)} node[above left, align=right, font=\footnotesize]{stationary \\ (local max)} ;
  89    \addplot[mark=*, orange] coordinates {(8,-400)} node[below, align=left, font=\footnotesize]{stationary \\ (local min)} ;
  90  \end{axis}
  91\end{tikzpicture}\\
  92\begin{tikzpicture}
  93  \begin{axis}[enlargelimits=true, xmax=3.5, ticks=none, axis lines=middle]
  94    \addplot[color=blue, smooth, thick] gnuplot [domain=0.74:3,unbounded coords=jump,samples=500] {(x-2)^3+2} node [black, pos=0.9, left] {\(f(x)\)};
  95    \addplot[color=darkgray, dashed, smooth, thick] gnuplot [domain=1:3,unbounded coords=jump,samples=500] {3*(x-2)^2} node [black, pos=0.9, right] {\(f^\prime(x)\)};
  96    \addplot[mark=*, purple] coordinates {(2,2)} node[below right, align=left, font=\footnotesize]{stationary \\ inflection} ;
  97  \end{axis}
  98\end{tikzpicture}
  99
 100\include{../spec/calculus-rules}
 101
 102\end{document}